Hamiltonian path

This article is about the overall graph theory concept of a Hamiltonian path. For the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph, see Hamiltonian path problem.

However, despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman, who, in particular, gave an example of a polyhedron without Hamiltonian cycles.[1]

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A Hamiltonian path or traceable path is a path that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices.

A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (except for the vertex that is both the start and end, which is visited twice). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

Similar notions may be defined for directed graphs, where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices are connected with arrows and the edges traced "tail-to-head").

A Hamiltonian decomposition is an edge decomposition of a graph into Hamiltonian circuits.

An Eulerian graphG (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graphL(G), so the line graph of every Eulerian graph is Hamiltonian. Line graphs may have other Hamiltonian cycles that do not correspond to Euler tours, and in particular the line graph L(G) of every Hamiltonian graph G is itself Hamiltonian, regardless of whether the graph G is Eulerian.[4]

The number of different Hamiltonian cycles in a complete undirected graph on n vertices is (n − 1)! / 2 and in a complete directed graph on n vertices is (n − 1)!. These counts assume that cycles that are the same apart from their starting point are not counted separately.

The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and Øystein Ore. Both Dirac's and Ore's theorems can also be derived from Pósa's theorem (1962). Hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters.[5] Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has enough edges.

Bondy–Chvátal theorem operates on the closure cl(G) of a graph G with n vertices, obtained by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with degree(v) + degree(u) ≥ n until no more pairs with this property can be found.

Bondy–Chvátal theorem

A graph is Hamiltonian if and only if its closure is Hamiltonian.

As complete graphs are Hamiltonian, all graphs whose closure is complete are Hamiltonian, which is the content of the following earlier theorems by Dirac and Ore.

Dirac (1952)

A simple graph with n vertices (n ≥ 3) is Hamiltonian if every vertex has degree n / 2 or greater.

Ore (1960)

A graph with n vertices (n ≥ 3) is Hamiltonian if, for every pair of non-adjacent vertices, the sum of their degrees is n or greater (see Ore's theorem).

The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph.

Rahman-Kaykobad (2005)

A simple graph with n vertices has a Hamiltonian path if, for every non-adjacent vertex pairs the sum of their degrees and their shortest path length is greater than n.[6]

The above theorem can only recognize the existence of a Hamiltonian path in a graph and not a Hamiltonian Cycle. A similar sufficiency condition for Hamiltonian cycles is introduced by Kaykobad.[7] It follows-

A simplebiconnected graph with n vertices is Hamiltonian if, for every non-adjacent vertex pair the sum of their degrees and their shortest path length is greater than or equal to n+1 with strict inequality holding for at least one pair of vertices.